How can you make brown painted wood look like gray weathered wood?

Yes with paint. Usually Chalk Paint gives the Gray look most are looking for

Use white and grey paint with a dry brush technique. There are lots of videos on you tube to help you. Brown is a good base to start with, you're in good shape.

You could probably fake it to some extent by doing a dry brush painting technique with shades of gray and white and maybe a little tan. You would start with the darkest color and move to the lightest. Look up YouTube videos on dry brush painting.

https://www.hometalk.com/search/posts?filter=faux%20weathered%20wood

Hi! I think this may achieve the look you want:

https://angelamariemade.com/2016/11/how-to-create-a-weathered-wood-gray-finish/